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Geomview supports hyperbolic and spherical geometry as well as
Euclidean geometry. The three buttons at the bottom of the *Main*
panel are for setting the current geometry type.

In each of the three geometries, three models are supported:
*Virtual*, *Projective*, and *Conformal*. You can
change the current model with the *Model* browser on the
*Camera* panel. Each Geomview camera has its own model setting.

The default model is all three spaces is *Virtual*. This
corresponds to the camera being in the same space as, and moving under
the same set of transformations as, the geometry itself.

In Euclidean space *Virtual* is the most useful model.
The other models were implemented for hyperbolic and spherical
spaces and just happen to work in Eucldiean space as well:
*Projective* is the same as *Virtual* but by default
displays the unit sphere, and *Conformal* displays everything
inverted in the unit sphere.

In hyperbolic space, the *Projective* model setting gives a view
of the projective ball model of hyperbolic 3-space imbedded in Euclidean
space. The camera is initially outside the unit ball. In this model,
the camera moves by Euclidean motions and geometry moves by hyperbolic
motions. *Conformal* model is similar but shows the conformal
ball model instead.

In spherical space, the *Projective* model gives a view of half
of the 3-sphere imbedded in Euclidean 3-space. Spherical motions give
rise to projective transformations in the *Projective* model, and
to
Moebius
transformations in the *Conformal* model. In both of these
models the camera moves by Euclidean motions.

In *Projective* and *Conformal* models, the unit sphere is
drawn by default. You can turn it off and on at will using the
*Draw Sphere* button in the *Camera* panel. In the
*Conformal* model, polygons and edges are subdivided as necessary
to make them look curved. The parameters which determine this
subdivision can be set with the `set-conformal-refine`

GCL
command. See `(set-conformal-refine ...)`

.

There are several sample hyperbolic space objects in the
`data/geom/hyperbolic` subdirectory of the Geomview directory.
Likewise, the subdirectory `data/geom/spherical`
contains several sample spherical space objects.